1. In general, how is a barrel proofed? Actually, the entire gun is proofed, not just the barrels. In the UK, it is submitted (gunmaker may be required) to the Proof House (London or Birmingham) for "viewing" (observation and measurement) by highly skilled technicians. It is returned if it does not pass exacting standards of condition. If it passes "view," it is fired (used to be once but now is twice, BOMK)with highly controlled ammo. After firing, it is again "viewed" for any changes in dimensions of conditiion. In the USA, manufacturers are on the hook for any failures, so proof is by company standard. There is no standardized proof in USA.
2. Why do some doubleguns seem to have lots of proof marks? In the UK, guns go out of proof for certain maintenance and repair work (repeated or prolonged bore honing, rechambering, rib relaying, etc.). Also, many guns were proofed only for BP originally. You might find an old, well used gun that had original BP proofs, later nitro proofs, later yet reproof for bore enlargement, and still later yet reproof for chamberlengthening.
3. Do I need to get my guns reproofed? In particular, My Iver Johnson Supertrap, from which I have shot 25 rounds from both barrels, my Iver Johnson Special Trap, from which I have shot several rounds, my WWII Stevens 620, which was recently overhauled and test fired by the gunsmith. The mentioned guns don't seem like candidates for reproof. As USA guns, who would you get to reproof them? You could send them to the UK, but that does not seem like a good plan to me unless, of course, you have reason to be concerned about them. Then, a good USA gunsmith will be your best approach.
4. How do you know that the proof test round is not the straw that broke the camel's back and that the next round will be okay? Overload testing is a well developed engineering practice. Proof firing is calculated to take the gun near, but not past, the highest pressure it can be expected to withstand. As long as there are no dimensional or condition changes, the gun is unlikely to fail in normal use. Stress loading of steel is cumulative only in terms of fatigue - it is vastly unlikely that you will ever fire any barrel enough times to experience a fatigue failure (mulit-million cycles).
Thank you. Hope this helps some. If it is unclear, post back.
Lou